STATIC ANAlySIS OF plASTIC lImIT STATE OF THIN-wAllEd ...

TRANSACTIONS OF THE INSTITUTE OF AVIATION ISSN 0509-6669217, p. 14-32 , Warsaw 2011

STATIC ANAlySIS OF plASTIC lImIT STATE OF THIN-wAllEdbEAmS wITH OpEN CROSS-SECTIONS

SebaStIaN GaWłoWSkI

Institute of Aviation

Abstract

The purpose of the paper is to explore the potential applications of the Vlasov theory to

description of limit states of thin-walled beam sections. This study addresses static aspects only.

The diagrams of stresses in the elastic range (predicted in accordance with the Vlasov theory) are

utilised to obtain limit stress distributions.

The analysis covers several examples: torsion of I-section thin-walled beam, bending and torsion

of I-section profile and bending with torsion of a channel profile. In the case of the second example,

the engineering application is shown, too. The analysed diagrams of stresses in plastic hinges can

be used to develop the interaction surfaces formulae, which enable us to assess the load capacity

at collapse.

The finite element simulation (Abaqus) shows that values obtained by this method are lower

estimations. Computation verifies the adequacy of limit stress distributions predicted by using of

the Vlasov theory.

Keywords: thin-walled beam, the Vlasov theory, limit state, plastic hinge

INtroductIoN

underlying this study is the thin-walled beam theory formulated by Vlasov [1]. the graphtheory is applied to simplify the thin-walled beam description. this new approach to the Vla-sov theory is given by Piechnik [2]. the work [2] provides a complete solution to the problemof thin-walled beams mechanics in the elastic state. It explains why new elements were addedto the Vlasov theory of thin-walled beams e.g. cross-sectional forces (stress resultants): bimo-ment, the Vlasov torsional moment, the Saint-Venant torsional moment and shows the normaland shear stress formulae, determined by sectorial co-ordinate and sectorial static moment,respectively. the instructions, most helpful in construction of plots of stress distribution inplastic hinges in the cases of thin-walled beams, are provided in [3], [4] and [5]. additionally,the book [5] contains the steel profiles research data, which verify the considerations presen-ted in this study.

the practical application of thus derived distributions of stresses in plastic hinges involvesan algorithm, which enables the assessment of load capacity at collapse of thin-walled beamswith open cross-sections. the algorithm is presented in more detail elsewhere, [6] and [7].

15

the [6] contribution compares the three sets of analytical, experimental and computationaldata. the latter are obtained using the shell finite element simulations (abaqus)-[8].

this finite element method (fem) program is a good tool for further studies. the papers [9] and [10] introduce the „monitoring areas” concept.In this approach the limit diagrams of normal and shear stresses can be assumed to be more

accurate as shear stresses can linear vary across the thickness of thin-walled cross-section.

1. maIN aSSumPtIoNS

the present analysis is based on the Vlasov theory, simplified by introduction of some ele-ments of the graph theory. thin-walled beams are described by co-ordinate systems: the glo-bal system xyz and the local one xsn - fig. 1.

fig. 1. the quantities used in the description of thin-walled beams in accordance with the Vlasov theory

the first system allows for defining the beam’s geometry, whereas the components of stressand strain tensors are determined in the other one. the profile cross-section is reduced to themiddle line (dendrite) d. due to specific behaviour of thin-walled beams under the applied load(small stiffness and great warping of cross-sections), new quantities have to be introduced tocreate a right description of this group of structures.

on the profile cross-section plane-fig. 1, several characteristic points are located. the pointCS denotes the centre of shear. If applied forces pass the locus of CS, the thin-walled beam is onlybent (without torsion). Generally, this point will not coincide with the centre of gravity CG,which happen in the case of solid (thick-walled) beams. the point Q is an origin of the naturalparameter (co-ordinate) s. the location of CS and Q is found using a special procedure availablein the Vlasov thin-walled beam description. the vector ρ(s) indicates the locus of any point P(for any parameter s) on the middle line d. as it is shown in fig. 1, the local co-ordinate systemxsn can be defined at any point of the d line. the axis x of the local system is parallel to the beamcentre line-x-axis of the global system, whereas the axis s is tangent to the middle line and n-axis is perpendicular to xs plane. the global co-ordinate system axes y and z denote the princi-pal directions of the cross-section.

the Vlasov theory introduces two important functions, which determine the distributionsof normal and shear stresses on the cross-section: sectorial co-ordinate ω (s) and sectorial static moment Sω (s). the first quantity defines normal stresses and is written as follows:

(1) ( ) ( )s s sn

s

= ∫ d0

StatIc aNalySIS of lImIt State of thIN-Walled beamS WIth oPeN croSS-SectIoNS

the equation (1) represents a curvilinear integral over the middle line d, from the point Q (s=0)

to any considered point P. the second function Sω (s), is related to shear stresses and is obtai-

ned from (2).

(2)

the integral in equation (2) is calculated from the given point to the end of an analysed leg ofthe cross-section. the symbol δ(s) denotes the cross-section thickness, which can vary along themiddle line.

the solution to thin-walled beams mechanics problem is obtained by computing the �(x)(angle of beam rotation function) from the fundamental equation of the Vlasov theory (3).

(3)

where:

,

E-young modulus, v- Poisson ratio, Iω-sectorial moment of inertia, Is-torsional moment ofinertia, Mx(x) - function of total torsional moment (with respect to the centre of shear CS)

apart from the Vlasov theory, normal and shear stresses diagrams in plastic hinges are con-structed assuming that the material is elastic-perfectly-plastic. this model can be applied to si-mulations of behaviour of steel or aluminium - typical materials of thin-walled profiles. underthis assumption the stress distributions can be simplified using rectangles with one height forall parts of the cross-section. obviously, the value of this height must not exceed the yield stressmagnitude.

the presented algorithm for assessing load capacity at collapse is based on two assumptions.first, the huber-mises-hencky yield criterion is used to describe the limit state. In the case ofthin-walled beams this condition has the form (4).

(4)

the symbols σx and τxs denote normal and shear stress, respectively, whereas σY is a tensileyield stress value. Secondly, an assumption is made that the analysed profiles are these forwhich a plastic hinge appears before local instability. accordingly, only a specific group of thin-walled beams with limited slenderness ratio of cross-sections legs will be considered.

16 SebaStIaN GaWłoWSkI

2. torSIoN of I-SectIoN thIN-Walled beam

let us consider cantilever I-section profile under torsion, as shown in fig 2.

fig. 2. the case of twisted concentrated moment applied to I-section cantilever thin-walled beam

If the solid beam theory is applied, only one cross-sectional force (torsional moment) willexist-related to shear stresses. however the fixed end of profile is a warping restraint, whichcauses also normal stresses to appear. this phenomenon is adequately described by Vlasovtheory, which predicts the existence of new stress resultant (determined by normal stresses) -bimoment. the diagram of bimoment Bω and the remaining nonzero cross-sectional forces, in-cluding: the Vlasov torsional moment Mω and the Saint-Venant torsional moment Ms, are shownin fig. 3.

fig. 3. the diagrams of cross-sectional forces in the considered problem: a) bimoment, b) the Vlasov torsional moment, c) the Saint-Venant torsional moment.

the symbol L denotes a free end of beam abscissa (beam length)

the stress resultants distributions are determined by function α(x) and derived from equa-tion (3). the suitable derivative relations have the forms:

(5)

the limit state of typical rolled steel cantilever I-section profile (investigated by Strelbicka etal.-[5]) was simulated using the finite element analysis (program abaqus). the load applied tocomputational model of beam is equal to the limit magnitude, obtained experimetally. the loa-ding way in fem simulation, the experimental one (outlined in [5]) and that presented in fig.2are equivalent. the results (distributions of the huber-mises-hencky reduced stress) for par-ticular beam layers are shown in fig.4.

17StatIc aNalySIS of lImIt State of thIN-Walled beamS WIth oPeN croSS-SectIoNS


fig. 4. the distributions of the huber-mises-hencky reduced stress on different layers of profile. on the right bottom part of picture the views from free end of beam are presented.

both, fig. 3 and fig. 4 illustrate the same issue-effort of thin-walled I-section cantilever beam. In the case of fem analysis, orange areas denote the spots where plastic yield process is ac-

tive. In fact, the yield stress of beam web material is more than in the case of a flange, which isindicated by dark orange colour for the web.

In fig. 4 two characteristic yield zones are apparent. the first one extends from the free endand involves ¾ of the profile. In this case the plastic yield is located in exterior layers, which sug-gests that it is generated when shear stresses achieves its limit level, because shear stresseshave the biggest values at boundaries of the profile and become less in the interior (as in thecase of the Saint-Venant torsion). the neutral points of shear stresses are located near the mid-dle lines of cross-sections-fig. 4c. the existence of these yield areas was predicted by the Vla-sov theory. this form of beam effort is a consequence of great values of the Saint-Venanttorsional moment, related to shear stresses-fig. 3c, which are located in the same part of pro-file as the analysed yield zones.

the existence of second plastic yield area at the fixed end of the beam-fig. 4, was clearly de-scribed by the Vlasov theory, too. the locus (near warping restraints) means that this yieldingoccurs when normal stresses achieves its limit value. this fact is confirmed by fig. 3a

fig. 4 shows also that plastic zones pass through all layers of profile flange, causing a plastichinge to appear in this location. the specific form of plastic hinge (yielding of flanges and

non-loaded web) - fig. 4, is also observed during experiment-[5].the construction of normal and shear stresses diagrams for a plastic hinge will be based on

the elastic state analysis. fig. 3 shows, that limit stress distributions (cross-section x = 0) are determined only by two

cross-sectional forces: bω and mω. the value of the Saint-Venant torsional moment at the fixedend equals zero.

accordingly, stresses diagrams in plastic hinge are derived from formulas:

(6)

(7)

the equation (6) shows that in the elastic state the normal stress x distribution over thecross-section is determined by sectorial co-ordinate, whereas in accordance with (7) the dia-gram of shear stresses (related to moment Mω) is controlled by sectorial static moment (assu-ming that thickness is constant for each cross-section leg). functions ω(s) and Sω(s) for I-sectionare depicted in fig. 5.

fig. 5 (and fig. 3c-zero value of Ms at the fixed beam end) show that the Vlasov theory pre-dicts the existence of non-loaded web in the plastic hinge. this is shown by fem analysis - fig.4, which is a good verification of the analytic limit state model of an I-section thin-walled beam.

let us consider the sectorial co-ordinate diagram (normal stresses distribution in elasticrange)-fig. 5a. the plastic yield process begins when the maximum values of σx (at ends of flan-ges) achieve the limit magnitudes. acting on the assumption that material is perfect (no har-dening), the limit value must not be exceeded. the further loading generates the extension ofstress limit areas along flanges to the web position. When the loading process is complete, therectangular distributions of normal stresses appear-fig. 6a. the applied load-fig. 2 is respon-sible for asymmetry of this characteristic (the right top and left bottom parts of the section aretensioned whereas the right bottom and left top are compressed)-fig. 6a.


fig. 5. Plots of: a) sectorial co-ordinate, b) sectorial static moment, for bisymmetric I-section

fig.6. diagrams of stresses in plastic hinge, predicted in accordance with the Vlasov theory: a)normal, b) shear, for twisted cantilever I-section thin-walled beam

the ordinate of stresses is not equal to the magnitude of σY, because the plastic state is alsogenerated by shear component of the stress tensor. because only Mω exists (Ms=0), the value τxs

should be constant over the cross-section thickness and should be determined by Sω(s) only.let us consider the right top part of the cross-section-fig. 5b.

the sectorial static moment diagram is placed on positive side of n-axis in the local co-ordi-nate system, which implies that the Sω(s) values are positive for this location. the Mω magnitudeis also positive-fig. 3b, so in accordance with formula (7) the shear stresses at right top part ofsection should be negative. the appropriate distribution is depicted in fig. 6b-the shear stres-ses at analysed parts of the top flange are negative, because they have the opposite direction tos-axis in the local co-ordinate system. basing on the presented algorithm the remaining distri-butions can be found-fig. 6b. the shear stress value can be taken as constant for all points offlanges, because in plastic hinge (in the full plastic state) it is related to constant σx by equation(4), at each point.


that the assumed normal stresses distribution is correct, was confirmed by fem limit statesimulations. fig. 7 shows a map of σx for beam middle surface near the fixed end of I-sectiontwisted profile, obtained utilising shell elements.

fig. 7. Normal stress distribution in a plastic hinge obtained by fem (program abaqus) - twisted I-section cantilever profile - middle surface

the predicted diagram - fig. 6a is the same as this which bases on computation data, thatimplies that stress distributions in plastic hinge can be based on the Vlasov theory.

3. beNdING WIth torSIoN of I-SectIoN thIN-Walled beam

this case is often encountered in engineering practice. for example, such situation is recal-led in fig. 8.

fig. 8. the case of bent and twisted I-section cantilever thin-walled beam

five cross-sectional forces will appear in this case. the distributions of the three of thesealong the beam length: bimoment, the Vlasov torsional moment and the Saint-Venant torsional


moment, follow the same pattern as in the previous example - fig. 3. the remaining stress re-sultants: bending moment My and shear force Fz are well-known from solid beams theory.

the first varies linearly from zero at the free end of profile to the maximum value for x = 0,whereas shear force diagram is constant. It is readily apparent that the occurrence of bendingin the torsion case - fig. 3, does not change the most exerted cross-section locus.

hence the plastic hinge is again placed at the fixed end of the beam. the fact that My and Fz

exist, prompts us to develop stress limit diagrams using the following formulas:

(8)

(9)

the diagrams of new functions, which determine stress distributions in elastic state, areshown in fig. 9.

fig. 9. the diagrams of: a) z co-ordinate, b) static moment with respect to y-axis, for bisymmetric I-section

fig. 9a shows that the asymmetric normal stresses diagram - fig 6a, valid for torsion, will bedisturbed by a constant positive σx values in the top flange case and by constant negative quan-tities in the bottom flange case. that implies that tension should prevail over compression whenthe top flange is considered, whereas the reversed situation holds for the bottom flange.

combining the two diagrams - fig. 5a and fig. 9a leads to relocation of neutral points σx dis-tributions in flanges - fig. 6a.

additionally, normal stress due to bending in profile web suggests that limit diagram pre-sented in fig. 10a would be correct. the parameter u depends on the Bω/My ratio. the magni-tude σx is not equal to the tensile yield stress value, because shear stresses also exist.

the distribution of τxs in flanges - fig. 10b, is related to Mω (for typical cross-section sizes en-countered in practical applications the influence of shear force can be neglected). It appearsthat the τxs diagram for flanges is the same as in torsion - fig. 6b and fig. 10b.

the nonzero value of Fz causes the shear stresses to appear in web. for the analysed loadingexample - fig. 8, the magnitude of shear force is negative (the remaining cross-sectional forces


are positive). In accordance with equation (9) the sign of the shear stress must be opposite tothat of the static moment Sy(s). In the upper half of the web the Sy(s) values are positive whe-reas in the lower - negative. a thorough analysis of the local co-ordinate systems-fig. 5a revealsthe τxs distribution should be taken as constant for the entire web-fig. 10b. the magnitudes ofσx and τxs must satisfy formula (4), which guarantees the plastic state in each point of cross-sec-tion.

fig. 10. the distributions of: a) normal stresses, b) shear stresses, in a plastic hinge, for bent andtwisted cantilever I-section of a thin-walled beam (in accordance with the Vlasov theory)

the adequacy of the assumed diagram-fig. 10a, is confirmed by the shell element analysis(abaqus). the task in fig. 8 was modelled and all relevant conditions (boundary, load, etc.) dulyaccounted for. fem calculations were applied in simulations of the experiment described in [5].the value of force applied to the computational model of a beam is equal to the load capacityat collapse obtained from Strelbicka et al. and hence the limit state simulation is performed.

the distributions depicted in fig. 10a and fig. 11 are very similar, which is a proof that theVlasov theory can be employed to construct limit stress diagrams in the thin-walled beamscase.

fig. 11. Normal stress distribution in a plastic hinge obtained by fem (program abaqus)-bent and twisted I-section cantilever profile-middle surface


4. alGorIthm of determINatIoN of load caPacIty at collaPSe-calculatIoN Pro-cedure IN the caSe of thIN-Walled beamS

Predicted diagrams of stresses in plastic hinge can be utilised to formulate the analytic ap-proach, which enables the assessment of load capacity at collapse of thin-walled beams withopen cross-sections. Study presented below is the outline of this method for bent and twistedI-section cantilever profiles. let us consider the general case all cross-sectional forces are non-zero. accordingly, diagram in fig. 10a can be well used. In fig. 10b, however, the influence of theSaint-Venant moment should be taken into account. for positive Ms (when only the Saint-Venantmoment exist) the limit stress distribution for any cross-section leg is shown in fig. 12a.

fig. 12. the limit shear stress distributions: a) generated only by Ms moment for a single cross-section leg, b) generated by Ms, Mω and Fz stress resultants for I-section

in the case of beam bending and torsion

the assumed shear stresses must be oriented counter-clockwise. If this diagram is taken intoaccount in fig. 10b for each cross-section leg, we obtain the distribution given in fig. 12b. the diagram in the web-fig. 10b must be re-constructed, because only positive stress resul-tants can be considered.

the results of study are distributions shown in fig. 10a and fig. 12b. the application of pre-dicted diagrams in the derivation of interaction surface formula is illustrated by an algorithmin fig. 13.

the considered stresses distributions are utilised to develop equivalence relationships, givenin general form by expressions (10)-(14).

(10)

(11)

(12)


(13)

(14)

the equation (14) contains the derivative with respect to s. application of equations (10)-(14) in the case shown in fig. 10a and fig. 12b leads to the formulae expressing the dependence of cross-sectional forces on parameters: u, η, μ.

When these parameters are reduced, we derive algebraic equations governing normal σx andshear τxs stresses in plastic hinge-fig. 13.

fig. 13. the algorithm for deriving the interaction surface formulae

F s n z s Az xs

A

= ∫∫ ( , ) ( ) d


after solving the algebraic expressions, the calculated limit stresses are substituted to con-dition (4), which completes the derivation of the interaction surface formula (15).

(15)

where:

, , , ,

xo-abscissa of cross-section with a plastic hinge,h-height of I-section (measured between middle lines of flanges), b - width of flanges,δw-thickness of web, δf - thickness of flanges

this equation will be used for calculation of load capacity at collapse, in an example presen-ted in fig. 8. Numerical data are summarised below:

h=114 mm, b=74 mm, δw=5 mm, δf=6 mm, beam length-l=1.275 m, material constants-e =210 GPa, v = 0.3, σy =254.7 mPa, force eccentric-ec =6 cm.

the plastic hinge appears at the fixed end of the beam (xo = 0). the cross-section effort is determined by values of stress resultants. the cross-sectional forces are obtained from expressions (5) after solving a differential equation (3).

In this case boundary conditions for (3) are:

, -warping restraint, -absence of bimoment (16)

the magnitudes of stress resultants for xo=0 are: Bω=0.02944P, My=1.275P, Mω=0.06P, Ms=0, Fz=-P. In these relationships the [m] and [kN] units are valid.

Substituting the values of cross-sectional forces to formula (15), the limit magnitude of forceP (the load capacity at collapse) is calculated. for considered numerical data the desired valueis given in table 1 - first column.


tab 1. the set of load capacity at collapse values for I-section cantilever profile subjected to bending and torsion

Whether bimoment and bending moment (stress resultants related to σx) are of major im-portance is checked by calculation of load capacity at collapse for the case when the remainingnonzero cross-sectional forces: Mω, Fz are assumed to be zero. the result is shown in the secondcolumn of table 1. the difference is very small-the third column of table 1, hence in similar si-tuations (when a plastic hinge appears in the place where cross-sectional forces related to nor-mal stresses have extreme magnitudes and where the Saint-Venant torsional moment is equalzero) the influence of τxs on load capacity at collapse value can be omitted.

the example considered in this study was investigated by Strelbicka et al.-[5]. the discre-pancy between the tensile yield stresses of web and flanges materials was simulated in an ana-lytic approach by using of one weighted value (the web and flanges moments of inertia withrespect to y-axis had weight factors). the result of experiment is given in fourth column-table1. the load capacity at collapse is larger than that obtained analytically.

the method based on the Vlasov theory gives a more conservative assessment limit force,which is a major advantage. furthermore, the level of difference (set in the last column of table1) is acceptable. a certain underestimation is also confirmed by fem analysis whereby loadequal to the limit value (column 1 table 1) is applied to the beam model. the distribution ofequivalent plastic strains (the plastic zone range) is illustrated by fig. 14..

fig. 14. the equivalent plastic strains distribution for bent and twisted I-section cantilever profile (abaqus) - middle surface

the plastic yield areas do not spread over the entire cross-section, so indeed the load valuederived analytically will remain below load capacity at collapse.


the analytic, experimental and computational data display a high degree of correspondence,which confirms the accuracy of results obtained by a method based on the Vlasov theory.

5. the „moNItorING areaS” aPProach

let us consider fem simulations of limit state of a thin-walled cantilever channel beam - fig. 15. the load applied to a profile model is the same as in fig. 8. fig. 15a shows the normalstress distribution near the fixed end of the beam (where a plastic hinge appears) over the mid-dle surface, fig. 15b provides the same map for a boundary exterior layer (surface of profile).

fig. 15. the limit distributions of normal stresses: a) for middle surface, b) for boundary (exterior)surface, at fixed end (in plastic hinge) of bent end twisted cantilever channel profile


the tensile yield stress of the analysed material approaches 240000 kPa. fig. 15 reveals thatnormal stresses in a plastic hinge are nearing σY. these stress magnitudes are represented bythe first and the last ranges at maps legends - fig. 15. It is reasonable to suppose, recalling equa-tion (4), that values of shear stresses in plastic hinge are very small (and do not approach theshear yield stress τY). hence in the limit state the distributions of τxs are the same as in the elas-tic range. that is why predicted diagrams of elastic shear stress (linearly variable along theprofile thickness) in a plastic hinge better portrays the analysed situation.

for a cantilever thin-walled channel beam, subjected the load shown in fig. 8, the relevantdistributions are shown in fig. 16a and fig. 16b.

fig. 16. the diagrams of: a) elastic shear stresses related to ms, b) elastic shear stresses related to Mωand Fz, c) sectorial static moment, for channel profile in the case of bending and torsion

the first diagram-fig. 16a, applies to a positive Ms. this plot is characteristic of the Saint-Ve-nant torsion. the second diagram-fig. 16b, gives the Vlasov torsional moment Mω and shearforce Fz. the distributions in flanges are related with the two cross-sectional forces.

In accordance with formula (9), the positive Sω(s)-fig. 16c, in upper flange produces (for po-sitive Mω) the negative stresses (opposite to s-axis-fig. 16c).

In the case of a lower flange, we get the reverse situation. the influence of Fz in flanges re-veals the same features as Mω. In the web, however, shear force prevails over the Vlasov tor-sional moment work, so diagram with a constant turn (characterised of Fz) can be assumed.the stresses are directed from bottom to top-fig. 16b, as it was the case of the positive shearforce. the parameter c determines the difference between the web and flanges distributions.Stress plots in fig 16 do not vary along the length of profile legs, because they are basis for theanalytic method (enabling the assessment of load capacity at collapse) and therefore shouldnot be too complicated.

In the plastic hinge the condition (4) must be fulfilled at each point of the cross-section. a dia-gram of normal stresses in limit state linearly variable through the profile thickness would berequired, which complicates the problem. the good solution is provided by the „monitoringareas” idea. accordingly, the entire cross-section is divided into layers - fig. 17a.


fig. 17. the diagrams of: a) normal stresses in plastic hinge for bending and torsion - in accordance with „monitoring areas” approach, b) sectorial co-ordinate,

in the case of thin-walled channel beam. the application of the „monitoring areas” approach to theinvestigation of shear stresses indicated with dot line

the linear shear stress diagrams are substituted by stepped diagrams - fig. 17a - dot line, si-milar to normal stress distributions. for each layer the values of σx and τxs are constant. thenumber of used layers depends on the desired accuracy of load capacity at collapse calcula-tions. In accordance with equation (4), it is required that the greatest value of τxs in the exte-rior layer be accompanied by the smallest σx at the same place.

this explains the normal stress distributions through the cross-section thickness. Instead, the σx distribution along the length of cross-section legs is determined by ω(s) and z(s)

functions, as revealed by equation (8). When torsion plays the key role, the sectorial co-ordinatediagram prevails over z(s) co-ordinate and the distribution shown in fig. 17b is decisive. So,the diagram presented by fig. 17a describes normal stresses distribution in plastic hinge whereload eccentricity ec is large.

the characteristic phenomenon occurs when eccentricity ec is slightly larger than distanceay -fig. 17b. the impacts of positive Bω and My are equal. the first cross-sectional force is rela-ted to negative linear σx diagram in the upper half of the web-fig. 17b, whereas the positivebending moment generates the positive linear normal stress distributions (well-known fromthe solid beams theory) at an analysed place. the loading process is responsible for the balanceeffect in both σx diagrams at the web. this „non-loading” of the web must be taken into acco-unt when constructing the normal stress distribution in a plastic hinge for the given case.

the predicted diagram of σx for the limit state (when torsion (Bω) dominates)-fig. 17a, isconfirmed by fem (abaqus) simulations - fig. 15. the distributions along middle line and thro-ugh the cross-section thickness obtained analytically and by computation procedure are thesame. the greater values of normal stresses in middle surfaces of flanges (when compared toexterior layers) are presented by the first and last areas in the map legends - fig. 15. In the web,


the stress distribution through the cross-section thickness (fig. 17a) is reverse to that illustra-ted in fig. 15, because the directions of shear force in these two cases are opposite.

limit stress diagrams shown in fig. 16a, fig. 16b and fig. 17a can be utilised to derive inte-raction surface formula, which enables the assessment of load capacity at collapse. the algo-rithm is similar to that presented in fig. 13. however, the normal and shear stress states are notanalysed simultaneously, first we consider τxs and next the yield condition (4)is formulated foreach layer separately. finally, the σx problem (with normal stress magnitudes calculated in thetwo previous steps) is investigated.

the analytic approach to finding load capacity of thin-walled beams at collapse, based on„monitoring areas” approach, is particularly useful in the study of cases where the Saint-Ve-nant torsional moment (τxs) has major influence in limit state creation.

6. coNcluSIoNS

It is shown that the Vlasov theory can be well used for building the normal and shear stressdiagrams in plastic hinge for thin-walled beams and evaluating profiles exertions. basing onelastic stress distributions (consistent with sectorial co-ordinates and sectorial static momentsfunctions), the virtual expansion of yield zones leads to creation of proper limit stresses dia-grams. the fem (abaqus) simulation confirms the adequacy of predicted plastic hinges shapes.the computation data indicate that evaluation of thin-walled beam exertion (i.e. finding of theplastic hinge locus) through the analysis of cross-sectional forces diagrams (developed in ac-cordance with the Vlasov theory) produces good results. limit stress diagrams developed in thisstudy can be utilised to formulate interaction surface formulae, which enable assessment ofload capacity at collapse in the case of thin-walled beams.

the constructed analytic method affords us the thesis that we obtain the underestimated va-lues of limit load, which is verified by experimental and calculation data. It is reasonable to sup-pose that this approach can be used in design, because provides safe values of load capacity atcollapse. this method can be also treated as a tool for verification of new algorithms (algo-rithms for determination of limit load). It appears that in the cases when the Saint-Venant tor-sional moment does not contribute to the formation of a plastic hinge, the influence of shearstresses on the limit load can be omitted. conversely, when shear stresses are not negligible, the„monitoring areas” approach should be adopted to calculate more exact values of load capacityat collapse.

Acknowledgments

the author would like to express his gratitude to Professors Stefan Piechnik and ryszard b.Pęcherski for their permanent interest and many valuable suggestions during the work on thissubject.

bIblIoGraPhy

[1] Vlasov WZ. thin-walled elastic beams [in russian]. moscow: State Publishers of Physicaland mathematical literature, 1959.

[2] Piechnik S. thin-walled beams with open cross-sections [in Polish]. krakow: krakow university of technology, 2000.

[3] Życzkowski m. combined loadings in the theory of plasticity. Warsaw: PWN - Polish Scientific Publishers, 1981.


[4] Strelbicka aI. Investigation of the strength of thin-walled beams beyond the elastic limit[in russian]. kiev: academy of Science, ukrainian S.S.r., 1958.

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SebaStIaN GaWłoWSkI

ANAlIzA STATyCzNA STANU gRANICzNEgO plASTyCzNEgO pRęTów CIENkOśCIENNyCH

O pRzEkROjACH OTwARTyCH

Streszczenie

Praca przedstawia implementację teorii Własowa do opisu własności prętów cienkościennych

otwartych w stanie granicznym plastycznym. W szczególności zajęto się konstruowaniem roz-

kładów naprężeń normalnych i stycznych w przegubach plastycznych. Rozważania pokazują, iż

teoria Własowa stanowi idealne narzędzie do przypuszczania granicznych plastycznych rozkła-

dów naprężeń w przypadku rozpatrywanej grupy profili. Rozwinięciem analizy jest skonstruo-

wanie podejścia analitycznego służącego szacowaniu nośności granicznej prętów cienkościennych

otwartych. Ilustracją rozważań są znane z praktyki inżynierskiej przykłady liczbowe obejmujące

zginanie i skręcanie profili dwuteowych oraz ceowych. Poprawność otrzymanych wyników ana-

litycznych jest potwierdzona przez przytoczone rezultaty obliczeń numerycznych MES jak i do-

stępne dane doświadczalne. Na zakończenie zaprezentowano ideę „monitorowanych pól”, która

pozwala na precyzyjniejszy opis zachowania się prętów cienkościennych otwartych w stanie gra-

nicznym plastycznym.


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